166 Expectation of Moments of Frequency Distributions 
-n . • „ — ^ IX,.r +i.f +rf +rf 
fAi) /^rf^lJ-rf,f^,y^f^rf^ 
/, (M /*-y, M,y^ /^-y , M/y^ H'rf . 
\^ (?) r 1 
+ ^ „ „ ' L*''7,+'7_. ^'7:,+'7. '^'7.,+'7,: + '^'7.+'7. ^'7 +'7. ^'7.+'7« ' - J' 
/, (I!) M'y, /'•'y, A^'y., ^'■'/j r'>y. f^' f,. 
and, on the other hand 
„(') 
1 =Mri+r2+... + ,7 = Mr, 
(.>/) 2' -'-1 
2 = ^ + . ^'-f7,+'7J + ■ • • 
./ = i .yi = i j'j=./i + i 
2;-A- + l ■2i-lc+2 2/ 
^ = 1 ./■.=.7i+l .» = /«•-. -rl 
^2;+]. 2= .-,/^'7^'-'7+ . ^'7,+''j./''-f'7+'7J ^ ••• 
;+2 / + 3 2/ + 1 
+ S 2 ... ^ fJ.,.j^+r^^...+rj.^i,-[rj^+r,^...+r.,^- 
,^1 = 1 ./.>=,yi + i 7( = /i-i + i 
In the case in which 7-, = ?•,> = ?■.= ... = the coefficients H, . become Ra ,) 
and we get again formulae (2), (4) and (5) of the present chapter. 
(8) Let us agree to denote 
We have : 
Eu, = EU\, - 1 Mr; in^,.,/^;,,! + 2 /.,^. flrj E [ m' } - ... 
(1) (2) 
(//)''" 
,y, ('-2) 
+ (-iy-M^-i)n,;,. 
Using here the vahies found above for EYX^d , A'fn^.y^ find so on, and denoting 
by K^h' ^' •■■ '''t-h+? the sum of all products of / factors of type jj,i,^fM)i^...fij,^ 
possible, subject to the conditions that h^, ho, ... hi appear in order, that the sums 
contain not less than two summands chosen from the numbers ?■], ... r^_j^^^, and 
that the number of summands in //«,. is not less than in h^+i, we find that the 
coefficient of 1 /iV' in the development of 7i',,/, dfi etpuils : 
